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Bohr was born in 1887 to Christian Bohr, a professor of physiology, from a Lutheran background, and Ellen Adler Bohr, a woman from a wealthy Jewish family of local renown.[2] Harald had a close relationship with his elder brother, which The Times likened to that between Captain Cuttle and Captain Bunsby in Charles Dickens' Dombey and Son.[3]

In the 1930s Bohr was a leading critic of the anti-Semitic policies taking root in the German mathematical establishment, publishing an article criticising Ludwig Bieberbach's ideas in Berlingske Aften in 1934.[7]

Bohr was also an excellent football player. He had a long playing career with Akademisk Boldklub, making his debut as a 16-year-old in 1903.[8] During the 1905 season he played alongside his brother Niels, who was a goalkeeper. Harald was selected to play for the Danish national football team in the 1908 Summer Olympics, where football was an official event for the first time. Though a Danish side had played at the 1906 Intercalated Games, the opening match of the 1908 Olympic tournament was Denmark's first official international football match. Bohr scored two goals as Denmark beat the French "B" team 9–0.[9] In the next match, the semi-final, Bohr played in a 17–1 win against France, which remains an Olympic record to this day.[10] Denmark faced hosts Great Britain in the final, but lost 2–0, and Bohr won a silver medal.[2][5] After the Olympics he made one further appearance for the national team, in a 2–1 victory against an England amateur team in 1910.[11] His popularity as a footballer was such that when he defended his doctoral thesis the audience was reported as having more football fans than mathematicians.[12]

Bohr was known as an extremely capable academic teacher and the annual award for outstanding teaching at the University of Copenhagen is called the Harald, in honour of Harald Bohr. With Johannes Mollerup, Bohr wrote an influential four-volume textbook Lærebog i Matematisk Analyse (Textbook in mathematical analysis).[13]